Simulation of cold hard balls related to explosions

New simulation and theory work by a trio of researchers in Massachusetts …

It is amazing what one can do when all you have is hard balls. Starting with Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller's seminal paper introducing the Monte Carlo algorithm and molecular Monte Carlo simulations in 1953, simulations of hard spheres* have revealed that these simplistic objects can describe a wide variety of physical phenomena. From being capable of demonstrating solid-fluid phase equilibrium, to illustrating a slow decay of velocity correlations despite the completely elastic collisions, this simple model—where particles are assumed to be what amount to billiard balls—is trivial to simulate and provides researchers with a great deal of insight.

A paper set to be published in an upcoming edition of Physical Review Letters looks at another application of hard spheres: explosions. To model this the researchers, T. Antal from Harvard, and P. L. Krapivsky and S. Redner from Boston University, looked at what happens when a single moving particle slams into a stationary hard sphere gas. They likened this simulation experiment to "the initial “break shot” in an infinite billiard table." Starting with a collection of stationary hard spheres, one sphere was given a large initial velocity and the resulting dynamics were examined. Interestingly, this setup is physically the same as an explosion occurring in a simple gas at 0 K.

Generic image of a simulation of
different sized hard spheres

In an attempt to describe the results of their simulation, the trio used various theories to describe two of the natural characteristics of the resulting shockwave of such an idealized explosion. Using hydrodynamic theory, the pair derived equations for the number of collisions that have occurred some time t after the initial motion. Derived from a simple rate theory was an expression for the number of hard spheres moving after some time t.

What they derived from these basic theories/relationships is that both properties, the numbers of moving particles and total collisions, were described by power-law relationships to the elapsed time since the initial blast. Perhaps surprisingly, both terms in the power part of the power law were defined only by the dimension of the system under examination. Finally they found, through simple mathematical analysis, that even through the initial particle motion is directed; for systems containing more than two dimensions, the center of mass of the particles does not move.

To test their theoretical predictions, the trio performed a series of molecular dynamic simulations of hard spheres that are initially at rest. The theory predicted that in two dimensions the number of moving particles should should be proportional to t (N ~ t), whereas the total number of collisions up to a time t is proportional to t raised to the 3/2 (C ~ t3/2). Simulation results found that the number of collisions is proportional to 0.95±0.1, and the number of collisions proportional to 1.45±0.1 for reduced densities in the range of 0.1 to 0.45. For dimensions greater than two, the authors note "that the cascade becomes symmetrical about its center of mass in spite of the directional initial perturbation," further confirming their theoretical predictions.

In addition to studying such collisions on an infinite table, the researchers look at the similar situation where all the particles are initially located in only one half of the space. Here all particles were placed in two dimensions, but their x coordinates were all greater than zero. The initial impact occurred at left edge of the block of particles and a shockwave propagated through the block of particles in a symmetric manner as expected. However, in addition to the shockwave moving to the right, a small number of particles ultimately get ejected backwards and escape to the left. What they found what that after long times, all of the energy is contained in the left half of space. The particles that do not get ejected ultimately decay back to zero energy states.

This works represents another interesting application of hard spheres and computer modeling. Particles of this type are extremely simple to model and to handle in various theoretical frameworks which make them ideal for work on new simulation algorithms and testing new theories.

* This first paper simulated hard disks, not hard spheres due to the limited computing power of the MANIAC computer available at Los Alamos at the time.

Matt Ford / Matt is a contributing writer at Ars Technica, focusing on physics, astronomy, chemistry, mathematics, and engineering. When he's not writing, he works on realtime models of large-scale engineering systems.